To solve the problem of finding the limit $\lim_{x \to 0} f(x)$ using the graph provided, let's analyze the behavior of the function $f(x)$ as $x$ approaches 0 from both the left and the right.

Step 1: Examine the graph as $x$ approaches 0 from the left ($x \to 0^-$):

• As $x$ approaches 0 from the left (values less than 0), the graph indicates that the function $f(x)$ is approaching the value $1$.

Step 2: Examine the graph as $x$ approaches 0 from the right ($x \to 0^+$):

• As $x$ approaches 0 from the right (values greater than 0), the graph indicates that the function $f(x)$ is approaching the value $-2$.

Step 3: Determine the limit:

• The limit $\lim_{x \to 0} f(x)$ only exists if the left-hand limit ($\lim_{x \to 0^-} f(x)$) and the right-hand limit ($\lim_{x \to 0^+} f(x)$) are equal.
• Here, $\lim_{x \to 0^-} f(x) = 1$ and $\lim_{x \to 0^+} f(x) = -2$. Since these two limits are not equal, the overall limit $\lim_{x \to 0} f(x)$ does not exist.

Answer:

• The correct choice is: "does not exist because the left- and right-hand limits are not equal."

Would you like more details on this topic, or do you have any other questions?

Here are five related questions that could further enhance your understanding:

1. How do you evaluate the limit of a piecewise function at a point of discontinuity?
2. What is the significance of left-hand and right-hand limits in determining the overall limit at a point?
3. How can the concept of limits be applied to real-world problems?
4. What are some common types of discontinuities found in graphs of functions?
5. How can a graph help in understanding the behavior of a function near a point of interest?

Tip: Always check both the left-hand and right-hand limits when evaluating the limit at a specific point to determine if the overall limit exists.